#10159196

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#40578	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3916 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,5)(4,8)(6,10)(7,14)(9,15)(11,16)(12,13)(17,18,23,28)(19,27,22,24)(20,21,26,25)(29,30,31,32)(33,34)(35,38)(36,41)(37,46)(39,42)(40,45)(43,44)(47,48);(1,3,6,9)(2,5,10,15)(4,12,14,11)(7,16,8,13)(17,19,23,22)(18,24,28,27)(20,29,26,31)(21,32,25,30)(33,35)(34,40)(36,43)(37,44)(38,42)(39,45)(41,48)(46,47);(1,4,6,14)(2,7,10,8)(3,11,9,12)(5,13,15,16)(17,20,23,26)(18,25,28,21)(19,29,22,31)(24,30,27,32)(33,36,39,46)(34,41,42,37)(35,43,45,47)(38,44,40,48);(1,4)(2,8)(3,12)(5,13)(6,14)(7,10)(9,11)(15,16)(17,20)(18,25)(19,29)(21,28)(22,31)(23,26)(24,30)(27,32)(33,37)(34,36)(35,44)(38,47)(39,41)(40,43)(42,46)(45,48);(1,3)(2,5)(4,11)(6,9)(7,13)(8,16)(10,15)(12,14)(17,21,23,25)(18,26,28,20)(19,30,22,32)(24,29,27,31)(33,34)(35,40)(36,41)(37,46)(38,45)(39,42)(43,48)(44,47);(1,5)(2,9)(3,10)(4,13)(6,15)(7,12)(8,11)(14,16)(17,22,23,19)(18,27,28,24)(20,31,26,29)(21,30,25,32)(33,38,39,40)(34,35,42,45)(36,44,46,48)(37,47,41,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10066392);